This simple, elegant construction yields a square simply by using the steps to bisect a segment. Begin by using your compass to draw a circle, making sure to mark its center. With your straight edge, draw the diameter of the circle. This diameter is the segment you will bisect, which will produce a new segment perpendicular to the diameter. Extend that segment so that it touches the circle, and you now have four evenly spaced points on your circle. Connect them to make a square.
Perform the same steps to bisect all four sides of the square, and you will have the points you need to make an octagon!